Since their introduction a decade ago, optical tweezers have become indispensable tools for physical studies of macromolecular and biological systems. Formed by bringing a single laser beam to a tight focus, an optical tweezer exploits optical gradient forces to manipulate micrometer-sized objects. Optical tweezers have allowed scientists to probe the small forces that characterize the interactions of colloids, polymers and membranes, and to assemble small numbers of colloidal particles into mesoscopic structures. Conventional efforts each required only one or two optical tweezers. Extending these techniques to larger and more complex systems requires larger and more complex arrays of optical traps.
Related techniques for creating multiple simultaneous optical traps include the generalized phase contrast method, interferometric optical tweezers, and optical lattices. The latter two approaches involve interfering multiple beams in the volume of the sample, while the former might be considered a variant of holographic optical trapping. Interferometric techniques can cover larger areas than holographic techniques, but are substantially more limited in the types of intensity patterns that can be created. In particular, interferometric optical tweezers and optical lattices are limited to periodic structures.
A single laser beam brought to a focus with a strongly converging lens forms a type of optical trap widely known as an optical tweezer. In general, such a beam can be described by a wave function,ψ(r)=A(r)exp(iφ(r))  (1)
where A(r) is the amplitude profile and φ(r) is the phase at position r in a plane transverse to the optical axis.
A conventional optical tweezer is created from the TEM00 laser beam provided by a typical laser. Such a beam's wave fronts are planar and can be described by the uniform phase profile φ(r)=φ0. Bringing such a beam to a diffraction-limited focus with an appropriate focusing element, such as a microscope objective lens, transforms the beam into an optical tweezer. The position of the optical tweezer in the lens' focal plane is determined by the angle at which the team enters the lens' input pupil. Additionally, if the beam is diverging as it enters the input pupil, it comes to a focus and forms an optical tweezer downstream of the focal plane. Alternatively, if the beam is converging, it forms a trap upstream of the focal plane.
Multiple beams of light passing simultaneously through the lens input pupil yield multiple optical tweezers, each at a location determined by the angle of incidence arid degree of collimation at the input pupil. These beams form an interference pattern as they pass through the input pupil, whose amplitude and phase corrugations characterize the downstream trapping pattern. Imposing the same modulations on a single incident beam at the input pupil would yield the same pattern of traps, but without the need to create and direct a number of independent input beams. Such wave front modification can be performed by a type of diffractive optical element (DOE) commonly known as a hologram. Generally, the hologram or DOE encoding a particular pattern of optical traps can be calculated with a computer through a procedure known as computer-generated holography (CGH). Using CGH to create arbitrary configurations of multiple optical traps constitutes a new class of optical micromanipulation tools known as holographic optical tweezers (HOT), with manifold applications in the physical and biological sciences as well as in industry.
The efficacy of holographic optical tweezers is determined by the quality of the trap-forming DOE, which in turn reflects the performance of the numerical algorithms used in their computation. Previous studies have applied holograms calculated by simple linear superposition of the input fields or with variations on the classic Gerchberg-Saxton and Adaptive-Additive algorithms. Despite their general efficacy, these algorithms yield traps whose relative intensities can differ substantially from their design values, and typically result in undesirable “ghost” traps. These problems can become acute for complicated three dimensional trapping patterns, particularly when the same hologram also is used as a mode converter to project multifunctional arrays of optical traps.
The holograms used for holographic optical trapping typically operate only on the phase of the incident beam, and not its amplitude. Such phase-only holograms, also known as kinoforms, are far more efficient than amplitude-modulating holograms, which necessarily divert light away from the beam. They also are substantially easier to implement than fully complex holograms that would be required to create arbitrary superpositions at the input pupil. Indeed, sequences of kinoforms can be projected with a computer-addressed spatial light modulator (SLM) to create dynamic holographic optical tweezers.
General trapping patterns can still be achieved with kinoforms despite the loss of information that might be encoded in amplitude modulations because optical tweezers rely for their operation on intensity gradients and not local phase variations. However, it is still necessary to find a. pattern of phase shifts in the input plane that encodes the desired intensity pattern in the focal volume.